All Questions
3,560 questions
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Constant mean curvature hypersurface
Assume that $f:\mathbb{B}^2\to \mathbb{C}$ is a holomorphic function defined in the unit ball in $\mathbb{C}^2$. Let $u(z)=|f(z)|(1-|z|^2)$ and consider $\Sigma =\{z: u(z)=c\}$. It seems to me that if ...
- 63.9k
4
votes
1
answer
848
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Does $P_xP_y+Q_xQ_y=0 \implies$ "non-existence of limit cycle" for $P\partial_x+Q\partial_y$"? (Complex dilatation and limit cycle theory)
Let $X=P\partial_x+Q\partial_y$ be a vector field on the plane $\mathbb{R}^2$. Assume that we have :$$P_xP_y+Q_xQ_y=0$$ Does this imply that the vector field $X$ is a divergence-free vector field ...
CommunityBot
- 1
0
votes
0
answers
57
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Double-periodic functions with (possible) poles
Consider the set of double-periodic function $f:\mathbb C/(\mathbb Z+i \mathbb Z) \setminus \{z_0\} \to \mathbb C$, where $z_0$ is a fixed point inside $\mathbb C/(\mathbb Z+i \mathbb Z),$ that have a ...
1
vote
0
answers
39
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About Carleson measures on the Hardy space on the bidisc
I have been reading the paper "Carleson Measures in Hardy and Weighted Bergman Spaces of Polydiscs" by F. Jafari and there are a few things that going on that I am not entirely convinced of.
...
2
votes
0
answers
165
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Bounds of modular functions on the Ford circles
Assume a holomorphic function from a product of two upper half planes $Z: \mathbb{H}_+\times \mathbb{H}_+\rightarrow \mathbb{C}$ with an expansion of the form
$$
Z(\tau,\tau') = \sum_{(h,h')\in S} a_{...
- 21
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0
answers
27
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Conformal mapping which is $\alpha$-holder continuous for each $\alpha<1$
Recall that the Jordan curve $J\subset \mathbb{C}$ is called asymptotically conformal if
$$\text{$\max_{z\in J(a,b)}\frac{|a-z|+|z-b|}{|a-b|}\to 1$ as $a,b\in J, |a-b|\to 0$}$$
where $J(a,b)$ is the ...
- 207
1
vote
1
answer
190
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Generalisation of Paley–Wiener type results for unbounded sets
Do you know an unbounded open set $A\subset \mathbb{R}^d$, $d\geq 2$ with the following property: if some integrable function $f$ on $\mathbb{R}^d$ has its Fourier transform vanishing on $A^c$ and all ...
- 6,367
3
votes
1
answer
127
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Can doubly parabolic Blaschke product (BP) contained in another doubly parabolic BP?
Let $f:\mathbb{D}\rightarrow\mathbb{D}$ be a degree $d$ doubly parabolic Blaschke product with Denjoy-Wolff point at $z=1$. That is, $f(1) = 1$, $f'(1)=1$ and $f''(1)=0$.
Let $U \subset \mathbb{D}$ be ...
3
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4
answers
1k
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$Aut(\mathbb{CP}^n)$ [..especially $n=1$ and $n=2$..]
I am confused and curious about the meaning of the $Aut(\mathbb{CP}^n)$.
Is what is called the "linear automorphism group" of $\mathbb{CP}^n$ the same as $Aut(\mathbb{CP}^n)$? It somehow seems to me ...
- 2,858
29
votes
7
answers
7k
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Elementary proof of Riemann-Roch for compact Riemann surfaces
I am supposed to give a talk about the Riemann-Roch theorem to a seminar of first and second year graduate students. I want to do Riemann-Roch for compact Riemann surfaces, but I am open to perhaps ...
- 63.9k
0
votes
0
answers
66
views
Uniformization and constructive analytic continuation of Taylor-Maclaurin series
Context. In their paper, "Uniformization and Constructive Analytic Continuation of Taylor Series", Costin and Dunne present a constructive method to greatly increase the accuracy of a ...
- 63.9k
3
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0
answers
125
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Analytic differential equations with estimates
I recently came across the following theorem of Martineau (and others). Let $P$ be a polynomial of $n$ variables with complex coefficients. If we identify naturally powers of partial derivatives $\...
- 6,367
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1
answer
444
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Expectation of complex random variable
I am researching frequency offset estimation and ended up reading a paper "Cramer-Rao Lower Bound on Frequency Offset Estimation Error in OFDM Systems With Timing Error Feedback Compensation"...
CommunityBot
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4
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0
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160
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An unusual uniqueness property for entire functions
For given $q\in (0,1),$ coefficients $|c_k|\leq Cq^{k^2/3},$ and non-negative non-decreasing convergent sequences $\{a_k\}_{k=0}^\infty$ and $\{b_k\}_{k=0}^\infty$ satisfying $a_k\geqslant b_k,\;k=0,...
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1
answer
202
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Probability of a number being a bound for roots
Consider the polynomial $p(z)=\sum_0^na_iz^k$ where $a_n=1$ and $a_k \sim N(0,1)$, $k=0,1,2,\dotsc,n-1.$
What is the probability that 2 will be a bound of the roots of the polynomial? How can we find ...
CommunityBot
- 1
27
votes
8
answers
5k
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Conceptual algebraic proof that Grassmannian is closed in Plücker embedding
I'm planning lectures for my intro algebraic geometry course, and I noted something awkward that is coming up. We're starting projective varieties soon. Of course, we'll prove that projective maps are ...
- 12.9k
1
vote
0
answers
113
views
Are there any known statistics on the sign of the Stieltjes Constants?
The Stieltjes Constants $\gamma_n$ arise from considering the laurent series of the Riemann Zeta function at $s=1$
$$ \zeta(s) = \frac{1}{s-1} + \sum_{n=0}^{\infty} (-1)^n \frac{\gamma_n}{n!} (s-1)^n $...
- 2,689
2
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0
answers
179
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Analytic continuation of $\int_V (f(x_1,\cdots,x_n))^s dx_i$
Let $V$ be an $n$-dimensional simplex, let $f(\boldsymbol{x}) = f(x_1,\cdots,x_n)\in \mathbb{C}[x_1,\cdots,x_n]$ be a product of linear polynomials that is non-zero in interior of $V$. Also let $E(\...
- 528
4
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0
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206
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It is possible, without adding further hypotheses, to refine Rouche's theorem in order to obtain a finer localization of the zeros?
The title says it all: a now deleted question on the Mathematics Stackexchange asked more or less the same thing, and I answered by citing the work [1] of Wolfgang Tutschke, whose version of Rouche's ...
- 6,367
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0
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37
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Theta series of well-rounded lattices
I've started looking into well-rounded Euclidean lattices and I was interested in learning whether their theta series have any interesting properties, but haven't found much in terms of bibliography ...
- 223
1
vote
2
answers
418
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Global theory of holomorphic functions [closed]
I am trying to develop a theory explaining analytic continuation of a holomorphic function $f(z)$ defined on an open set $D \subset \mathbb{C}$. Recently, I was looking at the last chapter of Lars ...
- 6,367
13
votes
2
answers
801
views
For which rationals is this exponential sum bounded?
Given $x \in [0, 1]$, we denote by $e(x)$ the complex number $e^{2 \pi i x}$.
Can we characterise the set of rationals $x$ for which the sum
$$A_N(x)\, :=\, \sum_{n = 0}^N e(2^n x)$$
remains bounded ...
- 6,323
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0
answers
82
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A far reaching generalization of the WPT?
I encountered the the Nullstellensatz for Germs of Holomorphic Functions in Daniel Huybrechts' Complex Geometry: An Introduction, specifically Proposition 1.1.29
If $I\subset \mathcal{O}_{\mathbb{C}^...
7
votes
2
answers
186
views
Non-locally connected polynomial Julia sets
What are some examples of complex polynomials whose Julia sets are connected, but not locally?
In the book Complex Dynamics by Carleson and Gamelin, I found:
They seem to reference:
But what is a ...
- 6,581
13
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1
answer
291
views
Descriptive complexity of analytic continuation
Consider the set of complex power series
$$
f(z)=\sum_{n=0}^\infty a_nz^n
$$ that have radius of convergence $1$ and can be analytically continued to the neighborhood of some point on the unit circle. ...
- 7,545
4
votes
1
answer
488
views
Linear combinations of roots of unity
Suppose that $x_i \in [-1,1], i=0,n-1$ and consider the root of unity $\omega = \cos(2\pi/n)+i\sin(2\pi/n)$ for some $n \geq 2$. Consider complex numbers of the form
$$ z = \sum_{i=0}^{n-1} x_i \omega^...
- 47.6k
4
votes
1
answer
173
views
Is every Cantor set $C\subseteq\mathbb R^{\infty}$ the limit set of a Fuchsian group?
Is every Cantor set $C\subseteq\mathbb R^{\infty}$ the limit set of a Fuchsian group?
Let's recall the definitions. A Fuchsian group $G$ is a discrete subgroup of $\operatorname{PSL}(2,\mathbb R)=\...
- 63.9k
4
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1
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328
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Holomorphic homotopy conjecture
Let $X$ be a smooth projective variety over the complex numbers, and let $\text{Coh}(X)$ be the category of coherent sheaves on $X$. Consider the dg-category $\text{Perf}(X)$ of perfect complexes on $...
- 471
4
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1
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214
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Explicit expression for a function in number theory
In their paper "Moyenne de certains fonctions arithmétiques sur les entiers friables", Tenenbaum and Wu proved that for the case of the function $\beta$ which is the indicator function of ...
- 14.7k
-2
votes
1
answer
121
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Infinite sum related to Hurwitz Zeta
I want to evaluate the following sum:
\begin{equation}
\sum_{-\infty}^{\infty}\frac{(-1)^n}{(n+a)^2}
\end{equation}
Where $a\in\mathbb{R}$ is not an integer. Such is similar to $\zeta(2,a)$, but it ...
- 188k
1
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0
answers
106
views
The proposition associated with a set
Given a set $U$ and a set $A \subseteq U$, is there an accepted symbol for the proposition $p$ over the universe $U$ such that for each $x \in U$, $p(x)$ is the assertion that $x \in A$? (The symbol $...
- 19.7k
2
votes
2
answers
363
views
Size of $\zeta'(s)$ at its zeros
How large can the derivative of the Riemann zeta function be at its zeros?
More specifically, let $\rho$ be a zero of the zeta function with $\Im(\rho)\in (0,T]$. What can we say about $|\zeta'(\rho)|...
- 11.1k
7
votes
2
answers
484
views
Tsuchiya-Ueno-Yamada's proof that sheaves of conformal blocks are locally free
I'm referring to Tsuchiya-Ueno-Yamada's (TUY hereafter) celebrated paper Conformal Field Theory on Universal Family of Stable Curves with Gauge Symmetries. One of the main goals of their paper is to ...
- 24.1k
9
votes
1
answer
393
views
A hypergeometric series for $\Gamma(1/4)^4/\pi^3$
Sorry if this comes out of the blue. Looking at old notes of mine, I found the identity
$$\dfrac{\Gamma(1/4)^4}{\pi^3}=4+\sum_{n\ge0}\binom{2n+1}{n}^3\dfrac{1}{2^{6n+1}}\;.$$
I cannot remember how I ...
- 1,389
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0
answers
65
views
Rotations and bi-analytic functions
Are the bi-analytic functions $\partial^2_{\overline{z}} f=0$
invariant under rotations?
- 41
3
votes
1
answer
116
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Interpretations of analytic continuations of CDFs to complex probabilities
Are there notable cases where analytic continuations of cumulative distribution functions to complex arguments have a meaningful interpretation or are otherwise useful?
If a one dimensional CDF is ...
- 7,545
2
votes
1
answer
263
views
Complex (i.e., Imaginary) Probability
I’ve been doing some numerical approximation of probability distributions. For continuous $\operatorname{PDF}$s (or $\operatorname{CDF}$s) greater smoothness can be exploited to achieve more ...
- 7,545
4
votes
1
answer
257
views
Asymptotics of an entire function with real zeroes on the real line
Define $ F(x) := A x^m e^{B x} \prod_{k \geqslant 1} (1 - x/\alpha_k)e^{x/\alpha_k} $ where we suppose that $ \alpha_k \in \mathbb{R} $. This function is defined on the whole complex plane and is ...
- 91.8k
20
votes
4
answers
2k
views
PDF readers for presenting Math online
In the current situation it seems especially important to be able to present your mathematical results online in a way that your audience does not fall asleep in front of their screens. But I am ...
- 1
12
votes
1
answer
735
views
Parametrisations for null temperature functions: nonuniqueness of solutions to the heat equation
Disclaimer. I expect this is a highly open problem, but maybe I'm wrong and someone has come up with some answers besides those given here. In any case, all information appreciated, thanks!
Definition....
- 6,367
2
votes
2
answers
2k
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Inversion of Laurent series
For a power series $f(z) = \sum_{i=0}^{\infty} a_i z^i$ with $a_1$ nonzero, Lagrange's inversion formula gives an explicit way to compute the Taylor coefficients of the inverse function.
Is there any ...
- 10.5k
10
votes
1
answer
444
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Analytic continuation gives a covering space (and not just a local homeomorphism)
Let $\mathcal{G}$ be the space of germs of holomorphic functions defined on open subsets of $\mathbb{C}$, topologized in the usual way. There is a natural map $p\colon \mathcal{G} \rightarrow \mathbb{...
- 149k
21
votes
6
answers
1k
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What is the relationship between $\sum_{n=0}^\infty f(n) x^n$ and $-\sum_{n=1}^\infty f(-n) x^{-n}$?
Background
Taking a relatively arbitrary combination of exponential and polynomial terms, for instance
$$\sum_{n=0}^\infty \left(n^{2}\sin\left(n\right)+n\cos\left(3n-2\right)\right)\cos\left(5n+1\...
- 6,367
4
votes
0
answers
323
views
Monstrous moonshine, Dedekind eta function, and the hypergeometric function
I. Monstrous Moonshine
Let $q = e^{2\pi i\tau}$ and $\tau = \sqrt{-d}$ or $\tau = \frac{1+\sqrt{-d}}2$ for positive integer $d$. Given the Dedekind eta function $\eta(\tau)$, consider the known ...
- 12.6k
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0
answers
48
views
When inclusion between two Kobayshi hyperbolic manifolds is distance decreasing?
Suppose that $X$ and $Y$ are two Kobayshi hyperbolic complex-analytic manifolds such that $X \subset Y$. It is known $d_Y(x_1, x_2) \leq d_X(x_1, x_2)$ for all $x_1, x_2 \in X$. In other words, the ...
- 41
4
votes
0
answers
148
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Some questions on Hardy's spaces
In the paper http://www.numdam.org/item/CM_1976__33_3_261_0.pdf, the authors have asked in Page 285 whether the Hardy space $H^p$ embeds isometrically into the Hardy space $H^q$ for $1\leq q<p<...
3
votes
1
answer
229
views
Minimum of a subharmonic function
Let for $j=1,\dots, m$, $z_j$ be distinct points from the unit disk $|z|<1$ and let $$g(z)=-\sum_{k=1}^m \log \frac{(1-|z|^2)(1-|z_k|^2)}{|1-z\overline{z_k}|^2}.$$ It seems that $g$ has a unique ...
- 1,538
2
votes
1
answer
99
views
A question on Bloch functions
Let $\mathcal{B}(\Delta)$ be the space of Bloch functions in the unit disk $\Delta$. For any $f\in \mathcal{B}(\Delta)$, we define the Bloch norm by
$$
\|f\|_{\mathcal{B}}=\sup_{|z|<1}|f'(z)|(1-|z|^...
2
votes
0
answers
125
views
Sub Banach spaces (Banach algebras) of the disc algebra which are invariant under the differentiation operator
In this question the disc algebra $\mathcal{A}(\mathbb{D})$ is the Banach algebra of all holomorphic functions on the unit open disc $\mathbb{D} \subset \mathbb{C}$ which have a ...
- 3,738
8
votes
3
answers
617
views
Uniqueness of Neumann series
Let $f$ be an entire function. Then there exist numbers $a_0,a_1,\ldots$, independent of $z$, such that
$$f(z)=\sum_{n=0}^\infty a_n J_n(z),\quad \forall z\in\mathbb{C}$$
where $J_n$ is the Bessel ...
- 60.6k